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Name ______________________
Exercise 13 Joins Count
Objective: Get familiar with Joints Count Statistics
Task: You will examine how Joints Count Statistics is constructed step by
step in Microsoft Excel.
Steps:
1. From the instructor’s folder copy, “Ex13.xls” to your own folder.
2. Open “Ex13.xls”. Notice that there are three worksheets. Each
worksheet contains the calculation of Joints Count Statistics for a
pattern of positive, negative or zero autocorrelation.
3. Click on worksheet “Sheet 1”. You will notice that there are two matrices.
 What is the size of both matrices? ________ by _________.
4. The values of both matrices are either 1 or -1. You also notice at the
right side of the worksheet, there is a chart. Each cell of the chart actually
corresponds to the value of a cell in the 1st matrix (above). Black (B) for
1 and grey (W) for -1. This chart basically shows the pattern of a binary
value (1 or -1) over the space.
 What is your best guess of the distribution of the values? Positive,
negative or zero autocorrelation? ____________________________
5. Click on worksheet “Sheet 2”. You will notice that it is very similar to
Sheet 1 but with only one matrix.
 What is your best guess of the distribution of the values? Positive,
negative or zero autocorrelation? ____________________________
6. Click on worksheet “Sheet 3”. You will notice that it is very similar to
Sheet 1 and 2.
 What is your best guess of the distribution of the values? Positive,
negative or zero autocorrelation? ____________________________
7. Switch back to worksheet “Sheet 1”. In Cell M2, type in
=IF(AND(B2=-1,C2=-1),"WW",IF(AND(B2=1,C2=1),"BB","BW")).
This equation is used to compare the values in Cells B2 and C2 (two
neighboring cells in the same row). If both B2 and C2 are -1, fill M2 with
“WW”; if both are 1, fill with “BB”; otherwise (B2 = 1 and C2 = -1; or
B2 = -1 and C2 = 1), fill with “BW”.
 Does the order of the combination (1 & -1 or -1 & 1) matter?
_____________________
8. Copy Cell M2. Paste it to Cells M2:U11. This 10 by 9 matrix contains all
the cases of two neighboring cells along each row (basically, joins along
East-West direction).
 Why 10 by 9? __________________________________
9. In Cell M14, type in =IF(AND(B2=-1,B3=1),"WW",IF(AND(B2=1,B3=1),"BB","BW")). This equation is used to
compare the values in Cells B2 and B3 (two neighboring cells in the
same column). If both B2 and B3 are -1, fill M14 with “WW”; if both are
1, fill with “BB”; otherwise (B2 = 1 and B3 = -1; or B2 = -1 and B3 =
1), fill with “BW”.
10. Copy Cell M14. Paste it to Cells M14:V22. This 9 by 10 matrix
contains all the cases of two neighboring cells along each column
(basically, joins along North-South direction).
 Why 9 by 10 this time? __________________________________
11.
So you have just listed all the possible join cases for a 10 by 10 matrix.
 What neighbor function is used? Queen, Rook, or bishop? ________
 How many join cases are there totally? ______________________
12. Next step is to count the number of cases for BB, WW, and BW
respectively. In Cell N25, type =COUNTIF(M2:V22,"BB"). This
equation counts the number of “BB” cases.
 What equation should be used for “WW”? ___________________
 What equation should be used for “BW”? ___________________
13.
Apply the above equations to Cell N26 and N27 respectively.
 JBB? ________________________
 JWW? _______________________
 JBW? ________________________
14. In Cell N29, type equation =SUM(N25:N27). This is the total number
of join cases (disregard the join type)
 Result? ___________________
15.
Cell N30 is for calculating m, which is given by:
1 n
m   ki (ki  1)
2 i 1
Where ki is the number of joins to the ith area.
16. Since we consider only the Rook-case neighbors, k is very easy to
calculate. Just break all the cells down into 3 groups: cells at corners, at
edges, at centers).
 How many corner cells? ________
 How many neighbors for each corner cell? ______
 How many edge cells? ________
 How many neighbors for each edge cell? ______
 How many center cells? ________
 How many neighbors for each center cell? ______
17. Following the example in your textbook (Equation 7.27), fill N30 with
=0.5*(4*2+32*3*2+64*4*3).
18. In order to get the expected values for JBB, JWW, JBW, you also need
to know the probability (pB) of a cell being coded B (1) and the
probability (pW) of a cell being coded W (-1).
 What is the probability for a cell being coded B? ________
 What is the probability for a cell being coded W? ________
19. Fill =COUNTIF(B2:K11,"1")/100 in Cell V25. This is the probability
for a cell being coded B (1).
 What should be the equation used for the probability for a cell being
coded W (-1)? ________________________
20. Apply the above equation in V26. This is the probability for a cell being
coded W (-1).
21. Next step is to calculate the expected values for JBB, JWW, JBW using
the equations given in your textbook (Equation 7.24).
22. Fill =N29*V25^2 in Q25, =N29*V26^2 in Q26; and
=2*N29*V25*V26 in Q27.
 E(JBB)? ________________________
 E(JWW)? _______________________
 E(JBW)? ________________________
 Comparing the expected joins counts with the observed (actual) joins
counts, positive, negative or zero autocorrelation? _______________
 Why? ______________________________________
23. Next step is to construct z-scores (through standardization, we can
compare cases with different number of cells).
24. We need the expected standard deviation for calculating z-scores. To
calculate the expected standard deviation for BB, WW, BW by simply
replaces the actual values in the equations given in your textbook
(Equation 7.25).
25. Fill =SQRT(N29*V25^2+2*N30*V25^3-(N29+2*N30)*V25^4)
in S25;
26. Fill =SQRT(N29*V26^2+2*N30*V26^3-(N29+2*N30)*V26^4)
in S26;
27. Fill =SQRT(2*(N29+N30)*V25*V264*(N29+2*N30)*V25^2*V26^2) in S27.
 E(SBB)? ________________________
 E(SWW)? _______________________
 E(SBW)? ________________________
28. Following the equation in your textbook (Equation A.15, p. 389), zscores can be rather easily calculated.
29.
Fill =(N25-Q25)/S25 in Cell X3. This is the z-score for BB case.
30.
Fill =(N26-Q26)/S26 in Cell X4. This is the z-score for WW case.
31.
Fill =(N27-Q27)/S27 in Cell X5. This is the z-score for BW case.
 z-score for BB? ________________________
 z-score for WW? ________________________
 z-score for BW? ________________________
 Positive, negative or zero autocorrelation? ______________________
 Why? ______________________________________
32. Select cells M1:X30. Copy and paste the entire selection to the same
area in “Sheet 2” and “Sheet 3”.
33.
For “Sheet 2”.
 z-score for BB? ________________________
 z-score for WW? ________________________
 z-score for BW? ________________________
 Positive, negative or zero autocorrelation? ______________________
 Why? ______________________________________
34.
For “Sheet 3”.
 z-score for BB? ________________________
 z-score for WW? ________________________
 z-score for BW? ________________________
 Positive, negative or zero autocorrelation? ______________________
 Why? ______________________________________
35.
Change the name of each worksheet to reflect your conclusions.